Deducing Temperatures and Luminosities of Stars (and other objects…)

Review: Electromagnetic Radiation Increasing energy Gamma Rays Ultraviolet (UV) X Rays Visible Light Infrared (IR) Microwaves Radio waves 10-15 m 10-9 m 10-6 m 10-4 m 10-2 m 103 m Increasing wavelength EM radiation consists of regularly varying electric magnetic fields which can transport energy over vast distances. Physicists often speak of the “particle-wave duality” of EM radiation. Light can be considered as either particles (photons) or as waves, depending on how it is measured Includes all of the above varieties -- the only distinction between (for example) X-rays and radio waves is the wavelength. Electromagnetic radiation is everywhere around us. It is the light that we see, it is the heat that we feel, it is the UV rays that gives us sunburn, and it is the radio waves that transmit signals for radio and TVs. EM radiation can propagate through vacuum since it doesn’t need any medium to travel in, unlike sound. The speed of light through vacuum is constant through out the universe, and is measured at 3x108 meters per second, fast enough to circle around the earth 7.5 times in 1 second. Its properties demonstrate both wave-like nature (like interference) and particle-like nature (like photo-electric effect.)

Wavelength  Wavelength is the distance between two identical points on a wave. (It is referred to by the Greek letter  [lambda]) One way of describing light is by its wavelength. Wavelength is the distance between the two identical points on the wave. The wave must be steady (no change in the oscillation and no change in its velocity) for it be possible to measure the wavelength. Wavelength is also shorthanded to a Greek letter Lambda.

Frequency time unit of time The same exact wave can be described using its frequency. Frequency is defined as the number of cycles of the waves per unit time. In the case shown, the frequency would be 1.5, since there are exactly one and a half complete cycles of the wave in the given time. The frequency is inversely proportional to its wavelength. Frequency is denoted by Greek letter “nu”. Frequency is the number of wave cycles per unit of time that are registered at a given point in space. (referred to by Greek letter  [nu]) It is inversely proportional to wavelength.

Wavelength and Frequency Relation Wavelength is proportional to the wave velocity, v. Wavelength is inversely proportional to frequency. eg. AM radio wave has a long wavelength (~200 m), therefore it has a low frequency (~KHz range). In the case of EM radiation in a vacuum, the equation becomes Wavelength and frequency are related to one another by the wave’s velocity. Wavelength is proportional (wavelength increases if velocity increases), and wavelength is inversely proportional to frequency (wavelength decreases if frequency increases). An AM radio wave has a large wavelength, so it has a low frequency (compared to other EM radiation.) In the case of EM radiation, the velocity is the speed of light, denoted by c. the speed of light is as mentioned before, approximately 3x108 meters per second. Using algebra, one can solve for any one of the variables. c Where c is the speed of light (3 x 108 m/s)

Light as a Particle: Photons Photons are little “packets” of energy. Each photon’s energy is proportional to its frequency. Specifically, each photon’s energy is E = h Now the particle nature of EM radiation. These little packets of light is known as photons. These photons carry a certain energy which is related to its frequency. This energy is equal to Planck’s constant (h) multiplied by the frequency of the photon. By substituting “nu” with the equation in the previous slide, we can get the equivalent equation in terms of wavelength. Planck’s constant is 6.6261 x 10-34 joule second Energy = (Planck’s constant) x (frequency of photon)

The Planck function Every opaque object (a human, a planet, a star) radiates a characteristic spectrum of EM radiation spectrum (intensity of radiation as a function of wavelength) depends only on the object’s temperature This type of spectrum is called blackbody radiation ultraviolet visible infrared radio Intensity (W/m2) 0.1 1.0 10 100 1000 10000

Temperature dependence of blackbody radiation As temperature of an object increases: Peak of black body spectrum (Planck function) moves to shorter wavelengths (higher energies) Each unit area of object emits more energy (more photons) at all wavelengths

Wien’s Displacement Law Can calculate where the peak of the blackbody spectrum will lie for a given temperature from Wien’s Law: 5000/T Where  is in microns (10-6 m) and T is in degrees Kelvin (recall that human vision ranges from 400 to 700 nm, or 0.4 to 0.7 microns)

Colors of Stars The color of a star provides a strong indication of its temperature If a star is much cooler than 5,000 K, its spectrum peaks in the IR and it looks reddish It gives off more red light than blue light If a star is much hotter than 15,000 K, its spectrum peaks in the UV, and it looks blueish It gives off more blue light than red light

Betelguese and Rigel in Orion Betelgeuse: 3,000 K (a red supergiant) Rigel: 30,000 K (a blue supergiant)

Blackbody curves for stars at temperatures of Betelgeuse and Rigel

Luminosities of stars The sum of all the light emitted over all wavelengths is called a star’s luminosity luminosity can be measured in watts measure of star’s intrinsic brightness, as opposed to what we happen to see from Earth The hotter the star, the more light it gives off at all wavelengths, through each unit area of its surface luminosity is proportional to T4 so even a small increase in temperature makes a big increase in luminosity

Consider 2 stars of different T’s but with the same diameter

What about large & small stars of the same temperature? Luminosity goes like R2 where R is the radius of the star If two stars are at the same temperature but have different luminosities, then the more luminous star must be larger

How do we know that Betelgeuse is much, much bigger than Rigel? Rigel is about 10 times hotter than Betelgeuse Rigel gives off 104 (=10,000) times more energy per unit surface area than Betelgeuse But the two stars have about the same total luminosity therefore Betelguese must be about 102 (=100) times larger in radius than Rigel

So far we haven’t considered stellar distances... Two otherwise identical stars (same radius, same temperature => same luminosity) will still appear vastly different in brightness if their distances from Earth are different Reason: intensity of light inversely proportional to the square of the distance the light has to travel Light wave fronts from point sources are like the surfaces of expanding spheres

Stellar brightness differences as a tool rather than as a liability If one can somehow determine that 2 stars are identical, then their relative brightnesses translate to relative distances Example: the Sun and alpha Centauri spectra look very similar => temperatures, radii almost identical (T follows from Planck function, radius can be deduced by other means) => luminosities about the same difference in apparent magnitudes translates to relative distances Can check using the parallax distance to alpha Cen

The Hertsprung-Russell Diagram